We now set up the tools from the operator perturbation theory [1]} for further analysis of the randomized FPCA. For the operator \( \widehat{\mathcal {C} }_{XX}\) , its resolvent is defined as \(\mathcal {R}_{\widehat{\mathcal {C} }_{XX}}(\eta ): = ( \widehat{\mathcal {C} }_{XX}-\eta I)^{-1}\) , where \(I\)  is the identity mapping. Let \(\Gamma _R:=\lbrace \eta \in \mathbb {C}:\ \mathrm {dist}(\eta ,\,[\hat{\sigma }_R^2, \hat{\sigma }_1^2]) = g_R/2\rbrace \)  represent the boundary of a disk in a complex plane. Every point on \(\Gamma _R\)  has equal distance \(g_R/2=(\hat{\sigma }_{R}-\hat{\sigma }_{R+1})/2\)  to the interval \([\hat{\sigma }_R^2, \hat{\sigma }_1^2]\subset \mathbb {R}\)  on the positive real axis.
Then, the subspace projection operator can be expressed as a contour integration
\(\widehat{\mathcal {P} }_R = -\frac{1}{2\pi i} \oint _{\Gamma _R} \mathcal {R}_{\widehat{\mathcal {C} }_{XX}}(\eta ) \,\mathrm {d}\eta \) .
